What is the difference between Bone Layers and Bone Groups?
Bone Layers (technically Armature Layers) is a way to hide visibility of rig controls.  When accessed from the Property window, it's a bit confusing.  It's much more intuitive when seen in the "N" panel (see below)
https://docs.blender.org/manual/en/latest/animation/armatures/properties/skeleton.html#armature-layers


Bone Groups are a way to organize your rig controls by color (also allows selecting entire categories of bones).  Below, I've got IK controllers hilighted and you can see the color set associated with IK- RED for bone color, BLUE for selected bones (the left hand), LIGHT BLUE for active bones.
https://docs.blender.org/manual/en/latest/animation/armatures/properties/bone_groups.html